 Hi, I'm Zor. Welcome to Unizor Education. I'll talk today about squares. But before doing the squares, I would like to talk about the hierarchy of certain geometrical objects, which I have addressed so far, within the area of quadrangles. Now, we have talked about parallelograms, about rhombuses, about rectangles. So now it's the squares turn. But what's the relationship between them? Okay, so let's talk about the relationship. Within the whole set of quadrangles, I have defined a subset of parallelograms. Now, parallelograms are quadrangles, which have, in addition to being quadrangles, which means four sides, four vertices, they have certain additional characteristics which define them as parallelograms. Well, basically it's parallelism of opposite sides. Okay, then I was talking about rhombus, which is a parallelogram. So in as much as parallelogram is subset of quadrangles, rhombus is subset of parallelograms. In addition to whatever the characteristics parallelogram has, which is opposite sides are parallel, rhombus has additional characteristic. Opposites, all four sides should be congruent to each other. What does it mean? It means that all the properties which we have proven for parallelograms, all the theorems, they are true for rhombuses as well. In addition, rhombus has certain specifically related to the fact that it's rhombus theorems, which we have also proven. Now, parallel to rhombus, and this is really very important, it's parallel to rhombus because it's from parallelograms, there is a concept of rectangles. Now, rectangle is also parallelogram, but rhombus adds to parallelogram congruence of the sides. Rectangle adds to parallelogram congruence of angles. So it's different characteristics, that's why I put them on the same level. They are both derived from parallelograms. And here, it's very appropriate now to put a square into this particular picture. Square is here, and it derives properties from both. Rhombus and rectangle. So a square is simultaneously a rhombus and a rectangle, which means it's parallelogram which has all four sides equal and all four angles equal to each other, congruent to each other. That's what the square is. And properties of the square are all the properties which parallelogram has plus all the additional properties which rhombus has plus all the additional properties that rectangle has. All together, all these properties make up properties of the square. So I don't really have to prove anything extra, there are not much actually which can be proven about the square, over and beyond whatever has already been proven for rhombus and rectangle. But I will just state these certain properties which square inherits either from parallelogram or from additional properties of rhombus or from additional properties from rectangle. So all these three categories, properties of the parallelogram, properties of the rhombus and properties of the rectangle are all added together and square basically has all these properties. So what are these properties? I will be very brief because we have already started all of these. So first I will mention the properties which square inherits from the parallelogram. Okay, two angles of the parallelogram, so this is our square. So the first property which is inherited from the parallelogram is that two angles adjacent to one side are supplemental to each other. Now these two angles are supplemental. Now we know that the square has 90 degree angles, all of them. Well, 90 and 90 do supplement each other, so that's true. But the property itself isn't inherited from the parallelogram because parallelogram has these two angles supplemental to each other, this one and this one. So these are also supplemental. Alright, that's good. Let's move on. Now parallel sides of parallelogram are congruent. Now this is parallel to this and these are congruent in parallelograms. This is congruent to this and these are congruent sides of the parallelogram. Now we know that this is true and we don't really have to basically prove the theorem again, not only because it was already proven for parallelograms, but because a square is a rhombus and we know that rhombus by definition has all the sides congruent to each other. But anyway, this is the property from the parallelogram that I just mentioned. Opposite angles are congruent, opposite angles are this and this. Again, we know they are all equal to each other because it's a rectangle and rectangle has all interior angles 90 degrees. So this is congruent to this. But again, the property actually goes back to parallelograms. Distance between parallel lines is the same. So distance between this and this, measured along any perpendicular, will be exactly the same. Same thing in this direction, distance between these two parallel lines. We have proven it during the parallelograms presentation. Point of intersection of diagonals with parallelogram divides each diagonal into two congruent pieces. So in the parallelogram, these two pieces of a diagonal and these two pieces of a diagonal are congruent to each other. Now, for a rectangle, we have proven that diagonals are congruent to each other as well, which means these pieces are all the same. So it's not two and two, it's all four, the same type of pieces, the same size of pieces. Okay, sequentially connected midpoints form parallelogram. So if you sequentially connect, I'll use another color. If you sequentially connect these. Now in general, you will have parallelogram. Now in particular, we will show it a little bit further when we will examine the properties of rhombuses and rectangles. It will be actually a square, but that's later. So far from the fact that this is a parallelogram, we only derived that this is a parallelogram as well. Now let's go to specific properties of the rhombus, which are also applicable to the square because the square is kind of a rhombus. First, diagonals are perpendicular to each other. So in any rhombus, including the square because it's a rhombus, diagonals are perpendicular to each other. We have proven it for a rhombus, so we don't have to really prove anything for a square. It's just the quality which we have inherited and proved before. So diagonals are perpendicular to each other. Also diagonals bisect the interior angles of the square. So this is an angle bisector of this angle. Same thing as this. So all these four angles are bisected by diagonals. Okay. Each diagonal is axis of symmetry. So this diagonal is axis of symmetry in the sense that this side above this diagonal can be turned and it will coincide with this one. That's what axis of symmetry actually means. In particular, it means that this is perpendicular and this is perpendicular to this line. These pieces are equal to each other and that's why the points are symmetrical relative to this diagonal. And analogously through this diagonal as well. Sequentially connected midpoints of rhombus form a rectangle. Okay, so now since it's a rhombus, we see that this is not just any parallelogram as we have derived from the consideration of the parallelogram. But this is actually a rectangle because in a rhombus if you add, let me just draw a different rhombus, more narrow. If you connect midpoints, you will have rectangle. Alright, so this is a rectangle now. Next. Okay, these are basically the properties which we have inherited from the rhombus. Parallelogram and the rhombus. Now let's add something which we have inherited from the fact that the square is a rectangle. Okay, all interior angles are right angles, so all these are right angles because it's a rectangle by definition of the square. If at least one interior angle is the right angle, then all of them are right angles. Again, it was very simple to prove because in any parallelogram if one particular angle is the right angle then all others also are right angles. Diagonals are congruent to each other. Okay, in the rhombus if you remember diagonals were perpendicular to each other. So we got perpendicularity of these. But now, since it's a rectangle as well, in a rectangle diagonals are congruent to each other. So now for a square we have a property that diagonals are not only perpendicular to each other but also congruent to each other as well. And they cross in the middle point of both of them. So the sequentially connected midpoints of a side of a rectangle form a rhombus. This is a very interesting situation. Well, let me just remind you, if this is a rhombus and you take the midpoints you get a rectangle. Now, if this is a rectangle and you take midpoints and connect them you will have a rhombus. Well, etc., etc., you can insert more and more. So what's interesting is that since this square is a rhombus then midpoints form a rectangle. But since it's a rectangle midpoints form a rhombus. That's why the midpoints point rhombus and rectangle which is a square by definition. So midpoints of a square make up a square. Okay? Now, line drawn through midpoints of opposite side is axis of symmetry. Remember for the rhombus diagonal is axis of symmetry. For a rectangle the line which connects midpoints of opposite side is axis of symmetry. Let me just draw this type of rectangle. Now you see that this and this are axis of symmetry. Diagonals are not axis of symmetry. If I turn this into this it will be something like this, right? But considering that our square is simultaneously rhombus and rectangle. So what do we have? We have both diagonals and both lines connecting midpoints of opposite side are axis of symmetry. So we have one, two, three, four. We have four axis of symmetry for a square. Is that it? Yeah, that was the last one which I wanted to talk about. Alright, so as you see we have basically lots of different properties which squares do have. But they are inherited from properties of the parallelograms, properties of rectangles or properties of the rhombus. There might be probably some additional theorems which are very, very specific for a square which are not true for neither rhombuses or rectangles. But these are probably some obscure qualities which we don't want really to talk about right now. As far as the fundamental properties of the square they are all related to inherited properties from the rhombus and from the rectangle. Basically that's it for today. One more quadrangle which we have not yet considered. It's trapezoid. That will be the next lecture. And meanwhile, don't forget the Unison.com is waiting for you for challenging problems and for supervised, controlled, if you wish, educational process. I would like to encourage parents and supervisors to enroll their children on this website into a specific topic or specific lecture, a specific lesson or whatever and let them study it, take the exam. There is a score so the supervisor or a parent can actually examine the score and say, okay, that's passed, let's move on. Or you have to study again, repeat the exam until you will get the good results. Okay, that's it for today. Thank you very much and good luck.